Internal combustion engine with rotor having offset peripheral surface

ABSTRACT

A rotary engine where the rotor cavity has a peripheral inner surface having a peritrochoid configuration defined by a first eccentricity and the rotor has a peripheral outer surface having a peritrochoid inner envelope configuration defined by a second eccentricity larger than the first eccentricity. Also, a rotary engine where the rotor cavity has a peripheral inner surface having a peritrochoid configuration defined by an eccentricity, and a rotor with a peripheral outer surface between adjacent ones of the apex portions being inwardly offset from a peritrochoid inner envelope configuration defined by the eccentricity. The engine may have an expansion ratio with a value of at most 8. The rotary engine may be part of a compound engine system.

TECHNICAL FIELD

The application relates generally to rotary internal combustion enginesand, more particularly, to rotors for such engines.

BACKGROUND OF THE ART

Wankel rotary engines typically include recesses in the rotor flanks,and the recess volume is used to adjust the compression and expansionratios of the engine as well as the minimum volume available forcombustion. In some instances, the maximum desirable recess volume maylimit the available compression ratio, expansion ratio and/or minimumvolume available for combustion.

SUMMARY

In one aspect, there is provided a rotary engine comprising: a housingdefining a rotor cavity with a peripheral inner surface having aperitrochoid configuration defined by a first eccentricity e_(H); and arotor rotationally received in the rotor cavity, the rotor having aperipheral outer surface defining a plurality of circumferentiallyspaced apex portions each including an apex seal biased away from theperipheral outer surface and engaging the peripheral inner surface ofthe rotor cavity, the peripheral outer surface of the rotor having aperitrochoid inner envelope configuration defined by a secondeccentricity e_(R), the second eccentricity e_(R) being larger than thefirst eccentricity e_(H).

In another aspect, there is provided a compound engine systemcomprising: a rotary engine having a housing defining a rotor cavitywith a peripheral inner surface having a peritrochoid configurationdefined by an eccentricity e_(H), and a rotor rotationally received inthe rotor cavity, the rotor having a peripheral outer surface defining aplurality of circumferentially spaced apex portions each including anapex seal biased away from the peripheral outer surface and engaging theperipheral inner surface of the rotor cavity, the peripheral outersurface of the rotor between adjacent ones of the apex portions beinginwardly offset from a peritrochoid inner envelope configuration definedby the eccentricity e_(H); a compressor communicating with an inlet portof the rotary engine; and a turbine connected to an exhaust port of therotary engine.

In a further aspect, there is provided a rotary engine comprising: ahousing defining a rotor cavity with a peripheral inner surface having aperitrochoid configuration defined by an eccentricity e_(H); and a rotorrotationally received in the rotor cavity, the rotor having a peripheralouter surface defining a plurality of circumferentially spaced apexportions each including an apex seal biased away from the peripheralouter surface and engaging the peripheral inner surface of the rotorcavity, the peripheral outer surface of the rotor between adjacent onesof the apex portions being inwardly offset from a peritrochoid innerenvelope configuration defined by the eccentricity e_(H); wherein theengine has an expansion ratio r_(EXP) defined by

$\frac{V_{MAX} + V_{R}}{V_{MIN} + V_{R}},$

where V_(R) is a volume of any recess defined in the peripheral surfacebetween the adjacent ones of the apex portions, V_(MAX) is a maximumvolume of a chamber defined between the peripheral inner surface of therotor cavity and the peripheral outer surface of the rotor duringrotation of the rotor, and V_(MIN) is a minimum volume of the chamber;wherein the expansion ratio r_(EXP) has a value of at most 8.

DESCRIPTION OF THE DRAWINGS

Reference is now made to the accompanying figures in which:

FIG. 1 is a schematic cross-sectional view of a Wankel engine;

FIGS. 2A and 2B are schematic representations of the generation of thegeometry of a peripheral inner surface of a rotor cavity and of aperipheral outer surface of a rotor, respectively, of a Wankel engine inaccordance with the prior art;

FIG. 3 is a schematic cross-sectional view of part of a peripheralsurface of a rotor of a Wankel engine according to a particularembodiment, as compared with a prior art rotor shown in dotted lines;

FIG. 4 is a graphic of a relationship between a relative size of a rotorrecess and a relative size of a rotor eccentricity as a function ofmultiple combinations of expansion ratios and rotor cavity trochoidconstants, in accordance with particular embodiment; and

FIG. 5 is a schematic representation of a compound engine system inwhich a rotary engine having a rotor such as shown in FIG. 3 and/or acombination of geometric parameters such as illustrated by FIG. 4 can beused.

DETAILED DESCRIPTION

FIG. 1 illustrates an example of a rotary intermittent internalcombustion engine 12 of the type known as a Wankel engine. It isunderstood that the configuration of the engine 12, e.g. placement ofports, number and placement of seals, etc., may vary from that of theembodiment shown.

The engine 12 comprises a housing 32 defining a rotor cavity 33 having aprofile defining two lobes. The housing 32 has a peripheral wall 38extending between two axially spaced apart end walls 54 to enclose therotor cavity 33. A rotor 34 is received within the rotor cavity 33. Therotor 34 has a peripheral outer surface 35 defining threecircumferentially-spaced apex portions 36, and a generally triangularprofile with outwardly arched sides. The apex portions 36 are in sealingengagement with the peripheral inner surface 39 of the rotor cavity 33,defined by the peripheral wall 38 of the housing 32, to form andseparate three working chambers 40 of variable volume between the rotor34 and the housing 32.

Referring to FIG. 2A, in known Wankel engines the peripheral innersurface 39 of the rotor cavity 33 is shaped to correspond orsubstantially correspond to a peritrochoid, defined by the locus of thetip point P of an arm fixed on a revolving circle B when it rolls alongthe periphery of a base circle A. The center distance between therevolving and base circles is defined as the eccentricity e and thelength of the arm fixed on the revolving circle is defined as thegenerating radius R. In practice, the peripheral inner surface 39 of therotor cavity 33 often corresponds to a peritrochoid with a generatingradius R which has been moved outward in parallel by a constant amount a(not shown), which closely approximates that of a peritrochoid having agenerating radius of R+a.

Referring to FIG. 2B, in known Wankel engines the peripheral outersurface 35 of the rotor 34 is shaped to correspond or substantiallycorrespond to the inner envelope of the peritrochoid, defined by theinner line of the pattern drawn by the peritrochoid of the rotor cavity33 (i.e. same generating radius R and eccentricity e) fixed on thecircle A when it rolls along the inner surface of the stationary circleB. In practice, the peripheral outer surface 35 of the rotor 34 oftencorresponds to the inner envelope of the peritrochoid with thegenerating radius R which has been moved outward in parallel by aconstant amount a′ (not shown), which closely approximates that of theinner envelope of a peritrochoid having a generating radius of R+a′,where a′=a+a minimum clearance determined based on errors inmanufacturing, thermal deformation, bearing clearance, etc.

The ratio of the generating radius R to the eccentricity (R/e) defines atrochoid constant K, a parameter which may be used to characterize thegeometry of the rotor 34 and rotor cavity 33. Practical values of K areusually from 6 to 8, although other values can be used. Smaller valuesof K produce rotors having a periphery approaching a triangular shapewhereas larger values produce rotors having a periphery approaching acircle.

Referring back to FIG. 1, the rotor 34 is engaged to an eccentricportion 42 of an output shaft 16 to perform orbital revolutions withinthe rotor cavity 33. The output shaft 16 performs three rotations foreach orbital revolution of the rotor 34. The geometrical axis 44 of therotor 34 is parallel to the axis 46 of the housing 32, and offsettherefrom by a distance corresponding to the eccentricity e. During eachorbital revolution, each chamber 40 varies in volume and moves aroundthe rotor cavity 33 to undergo the four phases of intake, compression,expansion and exhaust.

An intake port 48 is provided through the peripheral wall 38 foradmitting compressed air into one of the working chambers 40. An exhaustport 50 is also provided through the peripheral wall 38 for discharge ofthe exhaust gases from the working chambers 40. Passages 52 for a sparkplug or other ignition mechanism, as well as for one or more fuelinjectors of a fuel injection system (not shown in FIG. 1) are alsoprovided through the peripheral wall 38. Alternately, the intake port48, the exhaust port 50 and/or the passages 52 may be provided throughthe end or side wall 54 of the housing. A subchamber (not shown) may beprovided in communication with the chambers 40, for pilot or preinjection of fuel for combustion.

For efficient operation the working chambers 40 are sealed byspring-loaded apex seals 56 extending from the peripheral outer surface35 of the rotor 34 at each apex portion 36 to engage the peripheralinner surface 39 of the rotor cavity 33, and spring-loaded face or gasseals 58 and end or corner seals 60 extending from the rotor 34 toengage the inner surface of the end walls 54. The rotor 34 also includesat least one spring-loaded oil seal ring 62 biased against the innersurface of the end wall 54 around the bearing for the rotor 34 on theshaft eccentric portion 42.

Referring to FIG. 3, the peripheral outer surface of a prior art rotor34′ having a geometry as detailed above in relation to FIG. 2B isillustrated in dotted lines. The peripheral outer surface 35′ usuallyincludes a recess 70 in each flank, i.e. in the peripheral outer surface35′ between adjacent ones of the apex portions 36. The inclusion of arecess 70 lowers the compression ratio of the engine to practicablevalues (which otherwise could be for example from 15:1 to 21:1 forvalues of K from 6 to 8), and allows a volume for the initial combustionsince it occurs when the chamber 40 is positioned at its minimum volume.

The expansion ratio r_(EXP) of a Wankel engine can be defined by:

$r_{EXP} = \frac{V_{MAX} + V_{R}}{V_{MIN} + V_{R}}$

where V_(R) is the volume of the recess 70 defined in the peripheralouter surface 35, 35′ in each flank of the rotor (if present), V_(MAX)is the maximum volume of the chamber 40 defined between the peripheralinner surface 39 and the peripheral outer surface 35, 35′ (i.e. withoutthe recess volume), and V_(MIN) is the minimum volume of that chamber40. The compression ratio is typically similar or identical to theexpansion ratio (e.g. Otto cycle) or lower than the expansion ratio(e.g. Miller cycle obtained by moving the intake port 48 closer to topdead center (TDC)).

The rotor 34 according to a particular embodiment shown in full lines inFIG. 3 achieves similar expansion and compression ratios than the priorart rotor 34′ shown in dotted lines by increasing the size of theminimum volume V_(MIN) and decreasing the size of the recess volumeV_(R). The recess 70 can accordingly be omitted (as shown) or a smallerrecess may be present. The increase in minimum volume is obtained byflattening the shape of the peripheral outer surface 35 of the rotor 34,using a suitable convex shape which is contained within the conventionalinner envelope of the peritrochoid (as shown by the peripheral outersurface 35′ in dotted lines). In the embodiment shown, the convex shapeof the peripheral outer surface 35 corresponds to that of theconventional inner envelope of the peritrochoid at the apex portions 36,and is offset inwardly from the conventional peritrochoid inner envelopebetween the apex portions 36 by a distance progressively increasing fromeach apex portion 36 toward the center of the flank.

In a particular embodiment, the offset peripheral outer surface 35 ofthe rotor 34 has a profile corresponding to a peritrochoid innerenvelope configuration (i.e. a profile corresponding or substantiallycorresponding to a peritrochoid inner envelope) having the same or asimilar radius R as the peritrochoid configuration of the rotor cavity33, but an eccentricity e_(R) greater than the eccentricity e_(H) of theperitrochoid configuration of the rotor cavity 33; by contrast,conventional rotary engines use the same eccentricity e for the rotorand rotor cavity. The size of the rotor eccentricity e_(R) relative tothe rotor cavity eccentricity e_(H) is represented by the followingratio:

$\frac{e_{R} - e_{H}}{e_{H}};$

in a particular embodiment, the ratio

$\frac{e_{R} - e_{H}}{e_{H}}$

has any of the following values: any non-zero value up to and including40%; larger than 0 and at most 30%; larger than 0 and at most 20%;larger than 0 and at most 10%; at least 10%; at least 20%; at least 30%;at least 10% and at most 40%; at least 20% and at most 40%; at least 30%and at most 40%. Other values are also possible, including values higherthan 40%.

Alternately, the peripheral outer surface 35 of the rotor 34 can haveany other suitable convex shape offset inwardly from the conventionalperitrochoid inner envelope. For example, the offset peripheral outersurface 35 of the rotor 34 can have a profile corresponding to aperitrochoid inner envelope configuration having a radius R differentfrom that of the peritrochoid configuration of the rotor cavity 33.

FIG. 4 provides examples of how the recess volume V_(R) is adjusted fora given ratio between the rotor eccentricity e_(R) and the rotor cavityeccentricity e_(H) and a given trochoid constant K_(H) (R/e_(H)) of therotor cavity 33, in order to obtain a desired expansion ratio r_(EXP)(and compression ratio). The graph shows the size of the recess volumerelative to the rotor displacement, as illustrated by ratio

$\frac{V_{R}}{V_{MAX} - V_{MIN}},$

as a function of the size of the rotor eccentricity relative to therotor cavity eccentricity, as illustrated by ratio

$\frac{e_{R} - e_{H}}{e_{H}},$

for the combinations of expansion ratio r_(EXP) and trochoid constant Klisted in Table 1 below:

TABLE 1 function r_(EXP) K_(H) (R/e_(H)) F₁ 8 6 F₂ 7.5 6.9 F₃ 7.5 7.75F₄ 6.5 6.9 F₅ 6.5 7.75 F₆ 5.5 8

In a particular embodiment, the engine 12 with the offset peripheralouter surface 35 of the rotor 34 has an expansion ratio r_(EXP) havingany of the following values: 8 or less; 7.5 or less; from 5.5 to 7.5;from 5 to 7.5; from 5.5 to 8; from 5 to 8; about 6.5 and/or a trochoidconstant K_(H) of the rotor cavity having any of the following values:at least 6; at most 8; from 6 to 8; about 6.9; about 7.75. For example,respective values for the trochoid constant K_(H) of the rotor cavityand for the expansion ration r_(EXP) of the engine 12 may correspond toany combination found in table 1. Other values are also possible. In aparticular embodiment, these expansion ratios are obtained withouthaving any recess in the rotor flank, i.e. with V_(R)=0. In a particularembodiment, the compression ratio is similar or identical to theexpansion ratio. In another embodiment (e.g. Miller cycle) thecompression ratio is lower than the expansion ratio.

In a particular embodiment, the engine 12 with the offset peripheralouter surface 35 has the same expansion ratio r_(EXP) as a similarengine with a prior art rotor 34′ having a geometry as detailed above inrelation to FIG. 2B and having a recess volume V_(R) of between 5% and15% of the displacement volume, at least 6% and at most 11% of thedisplacement volume, and/or about 8 to 10% of the displacement volume.The engine 12 with the offset peripheral outer surface 35 achieves thesame expansion ratio r_(EXP) as this engine with the prior art rotor34′, but with a lower recess volume V_(R) or without any recess beingdefined in the rotor flank.

The rotary engine 12 is particularly, although not exclusively, suitableto be used in a turbo compounded cycle, since the low compression ratioproviding for reduced performance of the rotary engine 12 allows moreenergy to be recovered in compounding. Accordingly, in a particularembodiment illustrated by FIG. 5, the rotary engine 12 is used in acompound engine system 10 where one or more rotary engines 12 drive acommon load connected to the output shaft 16. The fuel injector(s) ofthe engine 12, which in a particular embodiment are common rail fuelinjectors, communicate with a source 30 of Heavy fuel (e.g. diesel,kerosene (jet fuel), equivalent biofuel), and deliver the heavy fuelinto the engine 12 such that the combustion chamber is stratified with arich fuel-air mixture near the ignition source and a leaner mixtureelsewhere. The compound engine system 10 also includes a turbocharger18, including a compressor 20 and a turbine 22 which are drivinglyinterconnected by a shaft 24, with the compressor 20 of the turbocharger18 compressing the air before it enters the rotary engines(s) 12. Theexhaust flow from the rotary engine(s) 12 is supplied to a compoundturbine 26 in fluid communication therewith, also driving the commonload, for example connected to the output shaft 16 through anappropriate type of transmission 28. The exhaust flow from the firststage turbine 26 is supplied to the second stage turbine 22 of theturbocharger 18.

The compound engine system 10 may be as described in Lents et al.'s U.S.Pat. No. 7,753,036 issued Jul. 13, 2010 or as described in Julien etal.'s U.S. Pat. No. 7,775,044 issued Aug. 17, 2010, the entire contentsof both of which are incorporated by reference herein.

In a particular embodiment, the low compression ratio obtained throughthe configuration of the offset peripheral outer surface 35 for therotor 34 allows to boost the inlet pressure higher for a similar peakpressure and therefore have higher power density, while the unusedenergy is recovered in the expansion via the turbines 22, 26.

In a particular embodiment, the expansion ratio of the turbines 26, 22is selected such that the turbine section provides a power outputcorresponding to from 20% to 35% of the total power output of thecompound engine system 10. In a particular embodiment, this may beachieved by having an expansion ratio in the turbine section which issimilar to the boost compression pressure ratio, i.e. the compressionpressure ratio of the compressor 20.

The increased power output of the turbine section may provide increasedpower for a given air mass flow, which may result in a smaller, lighterand more efficient engine at a given power. The low volumetriccompression ratio of the rotary engine 12 may help heavy fuel (e.g.diesel, kerosene (jet fuel), equivalent biofuel) to remain at a pressurelow enough to prevent self-ignition which may help ensure that thecycles runs with direct injection with a source of ignition, may savestructural weight, and may reduce internal leakages.

Alternately, the rotary engine 12 may be used without the turbocharger18 and/or without the compound turbine 26, and with or without one ormore other rotary engine(s) 12 engaged to the same output shaft 16. In aparticular embodiment, the rotary engine 12 is used as or part of anautomobile engine. In a particular embodiment, the rotary engine 12 isused as or part of an aircraft engine (prime mover engine or APU).

In a particular embodiment, the shape of the rotor 34 having an offsetperipheral outer surface 35 without recesses provides for a geometrywhich is less complex to manufacture either by casting or machining,exposes less area to the combustion chamber 40 and/or prevents local“hot spots” caused by the increased material thickness which wouldotherwise have been present at the junctions defined at the ends of therecess. In a particular embodiment, the shape of the rotor 34 is closerto a triangular shape than a conventional Wankel rotor, and accordinglymay be smaller and/or lighter.

In a particular embodiment, rotor 34 having an offset peripheral outersurface 35 is symmetrical and as such can be installed on both sideswithout affecting the combustion chamber geometry with any enginerotation direction. The symmetrical geometry may lead to a reduction inthe number of different parts in the engine thus facilitating anincreased production volume of a single part, help prevent theinstallation of the rotor in the wrong position (which could occur witha non-symmetric rotor) and/or help reduce a potential for manufacturingerrors resulting from having different rotor configurations.

The above description is meant to be exemplary only, and one skilled inthe art will recognize that changes may be made to the embodimentsdescribed without departing from the scope of the invention disclosed.For example, although the rotor cavity has been described with two lobesand the rotor with three apex portions, alternately the Wankel enginecould have a rotor cavity with any suitable number n of lobes and therotor any number n+1 of apex portions. Still other modifications whichfall within the scope of the present invention will be apparent to thoseskilled in the art, in light of a review of this disclosure, and suchmodifications are intended to fall within the appended claims.

1. A rotary engine comprising: a housing defining a rotor cavity with aperipheral inner surface having a peritrochoid configuration defined bya first eccentricity e_(H); and a rotor rotationally received in therotor cavity, the rotor having a peripheral outer surface defining aplurality of circumferentially spaced apex portions each including anapex seal biased away from the peripheral outer surface and engaging theperipheral inner surface of the rotor cavity, the peripheral outersurface of the rotor having a peritrochoid inner envelope configurationdefined by a second eccentricity e_(R), the second eccentricity e_(R)being larger than the first eccentricity e_(H).
 2. The rotary engine asdefined in claim 1, wherein the peritrochoid configuration of theperipheral inner surface of the rotor cavity and the peritrochoid innerenvelope configuration of the peripheral outer surface of the rotor havea same generating radius.
 3. The rotary engine as defined in claim 1,wherein a ratio $\frac{e_{R} - e_{H}}{e_{H}}$ has a value or at most40%.
 4. The rotary engine as defined in claim 1, wherein the ratio$\frac{e_{R} - e_{H}}{e_{H}}$ has a value of at least 10%.
 5. The rotaryengine as defined in claim 1, wherein the rotary engine has an expansionratio of at most
 8. 6. The rotary engine as defined in claim 1, whereinthe expansion ratio has a value of at least
 5. 7. The rotary engine asdefined in claim 4, wherein the rotary engine has a compression ratiocorresponding to the expansion ratio.
 8. The rotary engine as defined inclaim 1, wherein a trochoid constant of the rotor cavity defined asR/e_(H) has a value of at least 6, R corresponding to a value of agenerating radius of the peritrochoid configuration.
 9. The rotaryengine as defined in claim 1, wherein a trochoid constant of the rotorcavity defined as R/e_(H) has a value of at most 8, R corresponding to avalue of a generating radius of the peritrochoid configuration.
 10. Therotary engine as defined in claim 1, wherein the engine has a rotorcavity trochoid constant K_(H) and an expansion ration r_(EXP) havingrespective values corresponding to any combination found in table
 1. 11.The rotary engine as defined in claim 1, wherein the peritrochoidconfiguration has two lobes and the rotor has three apex portions.
 12. Acompound engine system including the rotary engine as defined in claim1, the compound engine system further comprising: a compressorcommunicating with an inlet port of the rotary engine; and a turbineconnected to an exhaust port of the rotary engine.
 13. A compound enginesystem comprising: a rotary engine having a housing defining a rotorcavity with a peripheral inner surface having a peritrochoidconfiguration defined by an eccentricity e_(H), and a rotor rotationallyreceived in the rotor cavity, the rotor having a peripheral outersurface defining a plurality of circumferentially spaced apex portionseach including an apex seal biased away from the peripheral outersurface and engaging the peripheral inner surface of the rotor cavity,the peripheral outer surface of the rotor between adjacent ones of theapex portions being inwardly offset from a peritrochoid inner envelopeconfiguration defined by the eccentricity e_(H;) a compressorcommunicating with an inlet port of the rotary engine; and a turbineconnected to an exhaust port of the rotary engine.
 14. The compoundengine system as defined in claim 13, wherein an expansion ratio r_(EXP)of the rotary engine is defined by$\frac{V_{MAX} + V_{R}}{V_{MIN} + V_{R}},$ where V_(R) is a volume ofany recess defined in the peripheral surface between the adjacent onesof the apex portions, V_(MAX) is a maximum volume of a chamber definedbetween the peripheral inner surface of the rotor cavity and theperipheral outer surface of the rotor during rotation of the rotor, andV_(MIN) is a minimum volume of the chamber, the expansion ratio having avalue of at most
 8. 15. The compound engine system as defined in claim14, wherein V_(R) has a value of
 0. 16. The compound engine system asdefined in claim 14, wherein the expansion ratio has a value of at least5.
 17. The compound engine system as defined in claim 14, wherein therotary engine has a compression ratio corresponding to the expansionratio.
 18. The compound engine system as defined in claim 14, whereinthe expansion ratio has a value of about 6.5.
 19. The compound enginesystem as defined in claim 13, wherein a trochoid constant of the rotorcavity defined as R/e_(H) has a value of at least 6, R corresponding toa value of a generating radius of the peritrochoid configuration. 20.The compound engine system as defined in claim 13, wherein a trochoidconstant of the rotor cavity defined as R/e_(H) has a value of at most8, R corresponding to a value of a generating radius of the peritrochoidconfiguration.
 21. The compound engine system as defined in claim 13,wherein respective values for a trochoid constant K_(H) of the rotorcavity and for an expansion ration r_(EXP) of the rotary enginecorrespond to any combination found in table
 1. 22. The compound enginesystem as defined in claim 13, wherein the peritrochoid configurationhas two lobes and the rotor has three apex portions.
 23. A rotary enginecomprising: a housing defining a rotor cavity with a peripheral innersurface having a peritrochoid configuration defined by an eccentricitye_(H); and a rotor rotationally received in the rotor cavity, the rotorhaving a peripheral outer surface defining a plurality ofcircumferentially spaced apex portions each including an apex sealbiased away from the peripheral outer surface and engaging theperipheral inner surface of the rotor cavity, the peripheral outersurface of the rotor between adjacent ones of the apex portions beinginwardly offset from a peritrochoid inner envelope configuration definedby the eccentricity e_(H); wherein the engine has an expansion ratior_(EXP) defined by $\frac{V_{MAX} + V_{R}}{V_{MIN} + V_{R}},$ whereV_(R) is a volume of any recess defined in the peripheral surfacebetween the adjacent ones of the apex portions, V_(MAX) is a maximumvolume of a chamber defined between the peripheral inner surface of therotor cavity and the peripheral outer surface of the rotor duringrotation of the rotor, and V_(MIN) is a minimum volume of the chamber;wherein the expansion ratio r_(EXP) has a value of at most 8.